A.R. Tusnin, M. Prokic. Behavior of symmetric steel I-sections under combined bending and torsion actions in inelastic range

Transcription

1 A.R. Tusnin, M. Prokic Behavior of smmetric steel Isections under combined bending and torsion actions in inelastic range

2 In European codes and Russian standards for design of steel structures, calculation of members is performed taking into account development of plastic deformations. Bearing capacit check in the current Russian Code for design of steel structures is performed b formula n N M M x B 1 An R c cxwxn,min R c cwn,min R c W n,minr c Where n, cx, c plastic shape coefficients (factors) for axial force and bending moment, which account development of plastic deformations in the cross section. The existing normative documents for the design of steel structures in Russia do not include coefficient taking into account the development of plastic deformations in warping torsion. Therefore, the purpose of the stud was to determine this factor in warping torsion. Primaril, smmetric Isections under the action of onl bimoment were examined. Plastic shape coefficient in warping torsion is defined as the ratio of plastic bimoment to elastic bimoment: c B pl B el Studies were carried out b Trahair N.S. and expressions of plastic bimoment for a number of profiles were obtained. B pl

3 h h For smmetric Ibeams the plastic bimoment can be determined b using simple method of analzing bearing capacit at full development of plastic deformations. Fig. 1. Deformations and sectorial stresses in a smmetric Isection under warping torsion b Fig. 2. Determination of plastic bimoment Sðàñ Sñæ Mz Af hbf/4 bf/4 h Af hbf/4 bf/4 Mz Plastic shape coefficient: c el bf Af bf h Bpl W, S pl, com. S, ten. 4 1,5 B W I / t b h,el f 3 2 f 24

4 h For monosmmetric Isection value of is defined analogousl. B pl Fig. 3. Deformations and sectorial stresses in a monosmmetric Isection under warping torsion b1 b2 2 b2 t2 2 In case, when 2, at the loss of bearing capacit larger flange will be full b1 t1 3 elastic. Constant value should be taken somewhat less than the theoretical, because the elastic core of section is preserved in the smaller flange. Based on this c 1,47.

5 h Recommended formula for bearing capacit check of Iprofiles in warping torsion, considering development of plastic deformations has the form: B 1, where c. c W R 1,47 c These studies were extended to analzing stressstrain state of a smmetric Isection under combined bending moment and bimoment actions in inelastic range. Fig. 4. Stressstrain state of Isection under the combined action of bending moment about the X axis and bimoment t f hw t f t w и X b f a

6 h Bending moment and Bimoment These studies were extended to analzing stressstrain state of a smmetric Isection under combined bending moment and bimoment actions in inelastic range. Fig. 4. Stressstrain state of Isection under the combined action of bending moment about the X axis and bimoment t f hw t f t w и X b f a Width a on normal stress diagram is eas to determine from equalit of external and internal bending moments: M M int. M Awh 4 M Aw The width a is equal: a. t h t h 4t f f f

7 Bimoment resisted b the section is: Bint ct f ( c a) h, where c ( bf a) / 2. Substituting value a defined above in the formula for c, and equating external bimoment with internal we obtain: M Aw bf h M Ah w B 0.5A (1) f h 4 2 2t f 8t f Analsis of expression (1) shown, that ultimate bimoment depends on the value of bending moment acting combined with bimoment. Calculations were performed, where relations of ultimate bimoment to plastic bimoment B/ B are depending on the ratio of acting moment to a plastic moment. pl M / M pl Two schemes for determining required ratio were given: 1 st scheme: from condition M B M R, so that B / B R с W / B c cw сw cwn 2 nd scheme: bimoment is determined b expression (1), which is modified given the fact that in RF norms development of plastic deformations in crosssection is limited. Therefore, in the region of neutral axis the elastic core is preserved, and bimoment is defined b: M M wpl bf h M M wpl B 0.5AfR h h 2 2t f R 2t f R where M wpl M pl Af Rh moment resisted b wall with the development of plastic deformations and a t w. n n pl c n pl

8 Table 1 Calculation of the combined moment and bimoment actions Section tpe h w, cm t w, cm b f, cm t f, cm h, cm Aw, cm Af, cm I t, cm I w, cm W, cm W w, cm GI t, kn/m EI w, kn/m k R, kn/cm Af/Aw с с w Мpl, кn m Мwpl, кn m Вpl, кn m Calculation of 1st scheme В/Вpl at moment В/Вpl at moment 0.2Мpl В/Вpl at moment 0.4Мpl В/Вpl at moment 0.6Мpl В/Вpl at moment 0.8Мpl В/Вpl at moment Мpl Calculation of 2nd scheme В/Вpl at moment В/Вpl at moment 0.2Мpl В/Вpl at moment 0.4Мpl В/Вpl at moment 0.6Мpl В/Вpl at moment 0.8Мpl В/Вpl at moment Мpl

9 B/Bpl Fig.5. The variation of ratio B/ B pl with M / M pl M/Mpl Scheme 1 Section 1 Section 2 Section 3 Section 4 Section 5 In the first scheme calculations, ratio varies linearl with. In the second scheme calculations relationship is nonlinear, at the same time bimoment value is noticeabl higher than in the first scheme. To assess the reliabilit of theoretical relationships, numerical studies of Isection profiles were performed in computing complex Nastran.

10 Numerical studies of Isection profiles in Nastran Fig. 6. Computational scheme of cantilever under bending moment Geometrical and phsical nonlinearit of sstem was taken into account. Figure 4 shows the stressstrain relationship: for stresses up to ield strength, equal to 240 MPa, dependence is linear with an elastic modulus MPa; further nearl horizontal line with a slight increase up to 250 MPa at relative strain of 0.3.

11 Fig. 5. The distribution of normal stresses over the section at M 0,4M pl and B 0,6B pl For each scheme of combined moment and bimoment actions, nonlinear analsis was carried out as long as rod kept its bearing capacit. In the first stage following combinations of moment and bimoment actions were included: 1 combination: M = 0 and B = B pl ; 2 combination: M = 0,2M pl and B = 0,8B pl ; 3 combination: M = 0,4M pl and B = 0,6B pl ; 4 combination: M = 0,6M pl and B = 0,4B pl ; 5 combination: M = 0,8M pl and B = 0,2B pl ; 6 combination: M = M pl and B = 0 ;

12 Calculations shown that in 3, 4 and 5 combination, difference between the ultimate load (consisting of the bending moment acting jointl with bimoment) and applied load was reaching 14%. At the second stage bimoment value was adjusted so that bearing capacit was provided at full load for each of the combinations. Relations of internal forces, obtained b different methods, which ensured bearing capacit, are shown in Table 2. Relations of internal forces which ensured bearing capacit М/Mpl В/Вpl 1st option Table 2 2nd option Table 2 The numerical calculation Analsis of numerical results showed that the bearing capacit of Isection profile, taking into account the development of plastic deformations, is significantl less than the bearing capacit obtained, both theoreticall (option 2), and using the procedure similar to normative (option 1). In view of this, for practical calculations normative procedure needs to be clarified. Studies found that in bearing capacit check, a coefficient c ω needs to be changed so that over the entire range of M and B ultimate bearing capacit is preserved.

13 Conclusions Coefficient should be changed when changing the ratio of M M pl. Table 4 Recommended values of c ω М/Mpl В/Вpl Intermediate values of coefficient are determined b linear interpolation. Final check of smmetric Isection profile bearing capacit should be carried out according to formula: M B 1 cw R с W R n c n c where the coefficient c is determined b the current regulations, coefficient c ω b Table 4.

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